Absolute convergence of double Walsh–Fourier series and related results

We consider the double Walsh orthonormal system on the unit square , where { w m ( x )} is the ordinary Walsh system on the unit interval in the Paley enumeration. Our aim is to give sufficient conditions for the absolute convergence of the double Walsh–Fourier series of a function for some 1< p...

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Bibliographic Details
Published in:Acta mathematica Hungarica Vol. 131; no. 1-2; pp. 122 - 137
Main Authors: Móricz, Ferenc, Veres, Antal
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.04.2011
Subjects:
Mathematics
Mathematics and Statistics
42C10
dyadic modulus of continuity
double Walsh–Fourier series
26A16
dyadic
functions of
modulus of continuity
26A15
bounded fluctuation
absolute convergence
dyadic Lipschitz classes of functions in two variables
ISSN:0236-5294, 1588-2632
Online Access:Get full text
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Summary:We consider the double Walsh orthonormal system on the unit square , where { w m ( x )} is the ordinary Walsh system on the unit interval in the Paley enumeration. Our aim is to give sufficient conditions for the absolute convergence of the double Walsh–Fourier series of a function for some 1< p ≦2. More generally, we give best possible sufficient conditions for the finiteness of the double series where { a mn } is a given double sequence of nonnegative real numbers satisfying a mild assumption and 0< r <2. These sufficient conditions are formulated in terms of (either global or local) dyadic moduli of continuity of  f .
ISSN:0236-5294
1588-2632
DOI:10.1007/s10474-010-0065-z